6,278 research outputs found
Total subspaces in dual Banach spaces which are not norming
The main result: the dual of separable Banach space contains a total
subspace which is not norming over any infinite dimensional subspace of if
and only if has a nonquasireflexive quotient space with the strictly
singular quotient mapping
Subspaces containing biorthogonal functionals of bases of different types
The paper is devoted to two particular cases of the following general
problem. Let and be two types of bases in Banach spaces. Let a
Banach space has bases of both types and a subspace contains
the sequence of biorthogonal functionals of some -basis in . Does
contain a sequence of biorthogonal functionals of some -basis in
?
The following particular cases are considered:
=(Schauder bases, unconditional bases),
=(Nonlinear operational bases, linear operational bases).
The paper contains an investigation of some of the spaces constructed by
S.Belle\-not in ``The -sum of Banach spaces'', J. Funct. Anal. {\bf 48}
(1982), 95--106. (These spaces are used in some examples.
Free Internal Waves in Polytropic Atmospheres
Free internal waves in polytropic atmospheres are studied (polytropic
atmosphere is such one that the temperature of gas linearly depends on
altitude). We suppose gas to be ideal and incompressible. Also, we regard the
atmosphere of constant height with the "rigid lid" condition on its top to
filter internal waves. If temperature, density and pressure of such undisturbed
atmosphere do not depend on latitude and longitude then the internal waves are
harmonic with apriori unknown eigenfrequencies, the problem permits separation
of variables and reduces to the system of two ODE's. The first ODE (the
Laplace's tidal equation) is analyzed by author earlier. The second ODE
determines the vertical structure of the waves to be considered and has
analytical solution for polytropic atmospheres. There are 6 dimensionless
numbers, 2 for the Laplace's tidal equation and 4 for the vertical structure
equation. The solution is a countable set of the eigenfrequencies and
eigenfunctions of the vertical structure equation; every
eigenfrequency/eigenfunction corresponds to its own countable set of the
eigenfrequencies and eigenfunctions of the Laplace's tidal equation. Parametric
analysis of the problem has been done. It shows that there exists the solution
weakly depending on altitude-temperature variations and the atmosphere's height
for parameters modelling the Earth's troposphere (with the "rigid lid" between
the troposphere and the tropopause). The natural periods of internal waves have
been obtained for this case.Comment: 7 pages, 1 table, 1 figur
Topologies on the set of all subspaces of a banach space and related questions of banach space geometry
For a Banach space we shall denote the set of all closed subspaces of
by . In some kinds of problems it turned out to be useful to endow
with a topology. The main purpose of the present paper is to survey results on
two the most common topologies on
Test-space characterizations of some classes of Banach spaces
Let be a class of Banach spaces and let
be a set of metric spaces. We say that is a
set of {\it test-spaces} for if the following two conditions are
equivalent: (1) ; (2) The spaces admit uniformly bilipschitz embeddings into .
The first part of the paper is devoted to a simplification of the proof of
the following test-space characterization obtained in M.I. Ostrovskii
[Different forms of metric characterizations of classes of Banach spaces,
Houston J. Math., to appear]:
For each sequence of finite-dimensional Banach spaces
there is a sequence of finite connected unweighted
graphs with maximum degree 3 such that the following conditions on a Banach
space are equivalent:
(A) admits uniformly isomorphic embeddings of ;
(B) admits uniformly bilipschitz embeddings of .
The second part of the paper is devoted to the case when
is an increasing sequence of spaces. It is shown that in
this case the class of spaces given by (A) can be characterized using one
test-space, which can be chosen to be an infinite graph with maximum degree 3
A note on analytical representability of mappings inverse to integral operators
The condition onto pair () of function Banach spaces under which there
exists a integral operator with analytic kernel such that the
inverse mapping im does not belong to arbitrary a priori given
Borel (or Baire) class is found
On metric characterizations of the Radon-Nikod\'ym and related properties of Banach spaces
We find a class of metric structures which do not admit bilipschitz
embeddings into Banach spaces with the Radon-Nikod\'ym property. Our proof
relies on Chatterji's (1968) martingale characterization of the RNP and does
not use the Cheeger's (1999) metric differentiation theory. The class includes
the infinite diamond and both Laakso (2000) spaces. We also show that for each
of these structures there is a non-RNP Banach space which does not admit its
bilipschitz embedding.
We prove that a dual Banach space does not have the RNP if and only if it
admits a bilipschitz embedding of the infinite diamond.
The paper also contains related characterizations of reflexivity and the
infinite tree property
Radon-Nikod\'ym property and thick families of geodesics
Banach spaces without the Radon-Nikod\'ym property are characterized as
spaces containing bilipschitz images of thick families of geodesics defined as
follows. A family of geodesics joining points and in a metric space
is called {\it thick} if there is such that for every and
for any finite collection of points in the image of , there is
another -geodesic satisfying the conditions:
also passes through , and, possibly, has some more
common points with . On the other hand, there is a finite collection of
common points of and which contains and is
such that the sum of maximal deviations of the geodesics between these common
points is at least
Structure of total subspaces of dual Banach spaces
Let be a separable nonquasireflexive Banach space. Let be a Banach
space isomorphic to a subspace of . The paper is devoted to the following
questions: 1. Under what conditions does there exist an isomorphic embedding
such that subspace is total? 2. If such
embeddings exist, what are the possible orders of ?
Here we need to recall some definitions. For a subset we
denote the set of all limits of weak convergent sequences in by
. Inductively, for ordinal number we let
The least ordinal for which is called
the {\it order} of
Steady 1D Stationary Currents of Spherical Gas Layer
Spherical layer of ideal gas is considered. The layer is in the sphere's
gravity field. Existence possibility of steady 1D stationary currents of this
layer is studied. This problem simulates zonal winds taking place in the
atmospheres of some planets such as Venus, Titan, Jupiter and Saturn.Comment: 5 page
- β¦